I am a high school calculus student. I understand how to differentiate functions both with respect to their independent variable and with respect to other variables, but I am in question about the theory behind this.
Normally, when you differentiate a function with respect to its independent variable, for example $y=x^2$, you can use the exponent rule to obtain $\frac{\mathrm dy}{\mathrm dx} = 2x$. However, when differentiating $x^2$ with respect to time, we get $2x\frac{\mathrm dx}{\mathrm dt}$.
My teacher never really explained this, so I just tried to explain it myself. Please tell me if my thought process is accurate:
We started the year off with the limit definition of the derivative. Our teacher then showed us how there is a pattern with the limit definition of the derivative, which is where we came up with the exponent rule.
Therefore, the exponent rule is based off the limit definition of the derivative. One of the parameters of using the limit definition of the derivative is one must know how the variable your differentiating with respect to relates to the function (i.e. it is the independent variable of the function.)
Therefore, one can only use the exponent rule when your differentiating with respect to the independent variable of the function, because this is required of the limit definition of the derivative, and the exponent rule is based off of the limit definition of the derivative.
Is my reasoning sound? Is this why the derivative of $x^2$ w.r.t. time is $2x\frac{\mathrm dx}{\mathrm dt}$? Since we do not explicitly know how time relates to $x$, our function?